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1500 questions
74
votes
14 answers
What isn't a vector space?
I'm really confused about vector spaces. We're learning about them in Linear Algebra, and my book doesn't give good examples of what a vector space is. I understand sets and vectors, but I don't understand vector spaces. From the definitions they've…
Aleksandr Hovhannisyan
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74
votes
15 answers
Solving $DEF+FEF=GHH$, $KLM+KLM=NKL$, $ABC+ABC+ABC=BBB$
She visits third class and is $8$ years old (you can imagine how ashamed I felt when I said so to her). I helped her with lots of maths stuff today already but this is very unknowable for me. Sorry it's in German but I have translated it :) It's…
cnmesr
- 4,938
74
votes
4 answers
Difference between supremum and maximum
Referring to this lecture , I want to know what is the difference between supremum and maximum. It looks same as far as the lecture is concerned when it explains pointwise supremum and pointwise maximum
user31820
- 1,053
74
votes
12 answers
Why do we not have to prove definitions?
I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? It seems like some definitions should have a…
Zduff
- 4,420
74
votes
1 answer
What is the logic/rationale behind the vector cross product?
I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from the cosine rule. (Do correct me if I'm wrong.)…
Danxe
- 1,703
74
votes
2 answers
Continuity and the Axiom of Choice
In my introductory Analysis course, we learned two definitions of continuity. $(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ satisfies $f(z_n) \to f(a)$ . $(2)$ A function $f:E \to…
John Gowers
- 25,836
74
votes
2 answers
Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$
About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{c^\color{red}{3}+d^\color{red}{3}}.$$ For $r=p/q$ where…
mathlove
- 155,494
73
votes
9 answers
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$ I proved $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ by interpreting $AB$ as…
user365
73
votes
3 answers
Mathematical research of Pokémon
In competitive Pokémon-play, two players pick a team of six Pokémon out of the 718 available. These are picked independently, that is, player $A$ is unaware of player $B$'s choice of Pokémon. Some online servers let the players see the opponents…
Andrew Thompson
- 4,519
73
votes
3 answers
What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
I tried and got this $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an integer. $$\lim_{n\to\infty}n\sin(2\pi…
M. Amin
- 739
73
votes
11 answers
What is the difference between only if and iff?
I have read this question. I am now stuck with the difference between " if and only if " and " only if ". Please help me out. Thanks
user2857
73
votes
3 answers
Distribution of the product of two (or more) uniform random variables
Say $X_1, X_2, \dots, X_n \sim U(0,1)$ are independent and identically distributed (i.i.d.) uniform random variables on the interval $(0,1)$ . I am interested in finding the distribution of the product of $2$ , $3$ or more such random variables,…
lulu
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73
votes
5 answers
Matrices commute if and only if they share a common basis of eigenvectors?
I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement?
Yaroslav Bulatov
- 5,639
73
votes
0 answers
Dedekind Sum Congruences
For $a,b,c \in \mathbb{N}$, let $a^{\prime} = \gcd(b,c)$, $b^{\prime} = \gcd(a,c)$, $c^{\prime} = \gcd(a,b)$ and $d = a^{\prime} b^{\prime} c^{\prime}$. Define $\mathfrak{S}(a,b,c) = a^{\prime} \mathfrak{s}( \tfrac{bc}{d}, \tfrac{a}{b^{\prime}…
user02138
- 17,374
73
votes
11 answers
$A_4$ has no subgroup of order $6$?
Can a kind algebraist offer an improvement to this sketch of a proof? Show that $A_4$ has no subgroup of order 6. Note, $|A_4|= 4!/2 =12$. Suppose $A_4>H, |H|=6$. Then $|A_4/H| = [A_4:H]=2$. So $H \vartriangleleft A_4$ so consider the homomorphism…
Stephen Cox
- 869